Question

The water trough shown in the figure to the right is constructed with semicircular ends. Calculate its volume in gallons if the diameter of the end is 17 in. and the length of the trough is 6 ft. Round to the nearest TENTH(Hint: Be careful of units.)

Answer 1

the final volume in gallons is
`\( 35.4 \)` gallons.

The volume of the water trough with semicircular ends, a diameter of 17 inches, and a length of 6 feet is approximately 35.4 gallons when rounded to the nearest tenth. Here's the step-by-step calculation:

1. Calculate the radius of the semicircle (which is half the diameter):
`\( \frac{17 \text{ in}}{2} = 8.5 \text{ in} \).`

2. Convert the length from feet to inches for consistency:
`\( 6 \text{ ft} * 12 \frac{\text{in}}{\text{ft}} = 72 \text{ in} \).`

3. Calculate the cross-sectional area of a semicircle:
`\( A = (1)/(2) \pi r^2 \).`

4. Use the area to calculate the volume in cubic inches:
`\( V = A * \text{length} \).`

5. Convert the volume from cubic inches to gallons using the conversion factor (1 cubic inch = 0.00432900431 gallons).

So, the final volume in gallons is
`\( 35.4 \)` gallons.

Answer 2

The volume of the trough is 141.5 gallons

SOLUTION

Problem Statement

The question gives us a trough with semi-circular bases with a diameter of 17 inches and a length of 6 feet. We are asked to calculate the volume of the trough in gallons.

Method

- Before we begin solving, we should make all the units uniform.

- 12 inches make 1 foot. Thus, we can say that:

`17\text{inches}=(17)/(12)ft`

- To find the volume of the trough, we can simply find the volume of a cylinder that was cut in half. We simply find the volume of the full cylinder and then divide the result by 2.

- When the volume has been calculated, we should convert from cubic feet to gallons. We use the following conversion:

`1cubic\text{ foot}\to7.48052\text{ gallons}`

- Thus, we can write solve the question using the following steps:

Step 1: Find the full volume of the cylinder.

Step 2: Find the volume of the trough.

Step 3: Convert the result from Step 2 from cubic feet to gallons

Implementation

Step 1: Find the full volume of the cylinder.

`\begin{gathered} \text{Volume of a cylinder }=\pi*((d)/(2))^2* h \\ \text{where,} \\ d=diameter \\ h=\text{height or length of the cylinder} \\ \\ V=\pi*((17)/(12))^2*6 \\ \\ V=37.83ft^3 \end{gathered}`

Step 2: Find the volume of the trough.

`\begin{gathered} \text{Volume of trough}=\frac{\text{Volume of cylinder}}{\text{2}} \\ \\ \therefore\text{Volume of trough}=(37.83)/(2)=18.915ft^3 \end{gathered}`

Step 3: Convert the result from Step 2 from cubic feet to gallons

`\begin{gathered} \text{ Using the following conversion:} \\ 1ft^3\to7.48052\text{gallons} \\ \therefore18.915ft^3\to(18.915ft^3)/(1ft^3)*7.48052gallons \\ \\ =141.494\text{gallons}\approx141.5\text{gallons (To the nearest Tenth)} \end{gathered}`

Final Answer

The volume of the trough is 141.5 gallons